U.S. Pat. No. 4,120,239 shows a strapping machine in which the strap is tensioned around the object by moving the strap laterally of its feed path and sequentially gripping the supply end of the strap, first to produce a low tension when the strap is moved laterally so as to remove the slack of the loop formed around the object, and then, finally, to grip the supply end with a desired force to produce a final tension so as to draw the strap tightly around the object. The apparatus of this subject application is an improvement on the strap tensioning apparatus of the patent.